Spatiotemporal patterns in a diffusive predator–prey system with nonlocal intraspecific prey competition

Abstract

We investigate the effect of nonlocal intraspecific prey competition on the spatiotemporal dynamics of a Holling-Tanner predator-prey model with diffusion. We first establish the criteria for Hopf, Turing, double-Hopf, and Turing-Hopf bifurcations, and determine the stable and unstable regions of the positive equilibrium. For Turing-Hopf bifurcation, by analyzing the normal form truncated to the third order, we derive that, with strong nonlocal interaction, the system exhibits the tristable phenomena, that is, the coexistence of a stable spatially nonhomogeneous periodic orbit and two nonconstant stable steady states, as well as the existence of periodic orbits with two spatial wave frequencies induced by the nonlocal interaction. The main analytical difficulty arises from the nonlocal interaction that prevents the direct application of formulas for the coefficients of the normal form. Biologically, the emerging spatiotemporal patterns suggest that the global intraspecific competition can promote the coexistence of the prey and predator by allowing the prey maintain a critical total population size, which may provide an alternative approach in explaining the group formation of some prey species under the risk of predation. Some coexistence patterns are destabilized by introducing another prey species with local intraspecific competition, leading to the

Spatially inhomogeneous bifurcating periodic solutions induced by nonlocal competition in a predator–prey system with additional food

“…Some scholars have studied the predator-prey models with nonlocal competition. [21][22][23][24] Britton 25 and Furter and Grinfeld 26 introduced the nonlocal competition effect by modifying the u K as 1 K ∫ Ω G(x, 𝑦)u(𝑦, t)d𝑦, with some kernel function G(x, 𝑦). Chen and Yu 22 studied a diffusive predator-prey model with nonlocal competition effect, including stability of positive constant steady state and Hopf bifurcation.…”

“…Since the resources is limited in nature, there is a competition within the population, and the competition is usually nonlocal. Some scholars have studied the predator–prey models with nonlocal competition 21–24 . Britton 25 and Furter and Grinfeld 26 introduced the nonlocal competition effect by modifying the $$$ \frac{u}{K} $$$ as $$$ \frac{1}{K}{\int}_{\Omega}G\left(x,y\right)u\left(y,t\right) dy $$$, with some kernel function $$$ G\left(x,y\right) $$$.…”

“…Chen and Yu 22 studied a diffusive predator–prey model with nonlocal competition effect, including stability of positive constant steady state and Hopf bifurcation. Geng et al 23 studied Hopf, Turing, double‐Hopf, and Turing–Hopf bifurcations of a diffusive predator–prey model with nonlocal competition. Liu et al 24 studied a diffusive predator–prey model with nonlocal competition and time delay.…”

Spatiotemporal dynamics in a delayed diffusive predator–prey system with nonlocal competition in prey and schooling behavior among predators

“…In [17], Wu and Song studied a diffusive predator-prey model with nonlocal effect and delay, and suggested that steady-state, Hopf, and steady-state Hopf bifurcations may occur. In [18], Geng et al studied Hopf, Turing, double-Hopf, and Turing-Hopf bifurcations of a diffusive predator-prey model with nonlocal competition. In [19][20][21][22], all the authors show that the nonlocal competition may induce stably spatially inhomogeneous bifurcating periodic solutions, which is different from the model without nonlocal competition.…”

Multi-stable spatial patterns and spatiotemporal staggered periodic patterns in a predator-taxis model with delay

“…Throughout the paper, N is the set of all positive integers, and N 0 = N∪{0}. As for the definitions of the mode-k 1 Turing bifurcation and mode-(k 1 , k 1 + 1) Turing-Turing bifurcation, as well as mode-k 2 Hopf bifurcation, mode-(k 1 , k 2 ) Turing-Hopf bifurcation, mode-(k 2 , k2 ) Hopf-Hopf bifurcation and so on, which will be mentioned later, the reader may refer to [7,13,19].…”